How can I have an entire island move when selecting a portion of it?
Enable Proportional Editing in the UV editor, setting it to Connected and Constant. Then after selecting just move the island as close to the relevant pixel you can, ensuring the Proportional Edit is mousewheeled out (down on windows) enough to move the entire island.
If you need to be perfect, I guess move a randon, probably still-to-be-unwrapped vertex to the cursor with Snap>Selected to Cursor. The select your island, and with Proportional editing above, Change "Type of Element to Snap to" to Vertex (icon next to the Magnet). Assuming Snap during Transform is off, G to move your island, and if you hold Control, your island should snap to the vertex at the cursor when you get near.
Just remember to to move the "spare" / "cursor" vertex out again by a combination of inverting selection or some such.